Algebraic structures in comodule categories over weak bialgebras

Abstract

For a bialgebra L coacting on a k-algebra A, a classical result states that A is a right L-comodule algebra if and only if A is an algebra in the monoidal category M L of right L-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras H. The category M H admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in M H are not canonically k-algebras. Nevertheless, we prove that there is an isomorphism between the category of right H-comodule algebras and the category of algebras in M H . We also recall and introduce the formulaic notion of H coacting on a k-coalgebra and on a Frobenius k-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra L to the corepresentation category of a weak bialgebra built from L (a "quantum transformation groupoid"), thereby creating weak quantum symmetries from ordinary quantum symmetries.